Switching Checkerboards

Abstract

In order to study $\mathbf{M}(R,C)$, the set of binary matrices with fixed row and column sums $R$ and $C$, we consider sub-matrices of the form $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, called positive and negative checkerboard respectively. We define an oriented graph of matrices $G(R,C)$ with vertex set $\mathbf{M}(R,C)$ and an arc from $\mathbf{A}$ to $\mathbf{A'}$ indicates you can reach $\mathbf{A'}$ by switching a negative checkerboard in $\mathbf{A}$ to positive. We show that $G(R,C)$ is a directed acyclic graph and identify classes of matrices which constitute unique sinks and sources of $G(R,C)$. Given $\mathbf{A},\mathbf{A'}\in\mathbf{M}(R,C)$, we give necessary conditions and sufficient conditions on $\mathbf{M}=\mathbf{A'}-\mathbf{A}$ for the existence of a directed path from $\mathbf{A}$ to $\mathbf{A'}$. We then consider the special case of $\mathbf{M}(\mathcal D)$, the set of adjacency matrices of graphs with fixed degree distribution $\mathcal D$. We define $G(\mathcal D)$ accordingly by switching negative checkerboards in symmetric pairs. We show that $Z_2$, an approximation of the spectral radius $\lambda_1$ based on the second Zagreb index, is non-decreasing along arcs of $G(\mathcal D)$. Also, $\ll$ reaches its maximum in $\mathbf{M}(\mathcal D)$ at a sink of $G(\mathcal D)$. We provide simulation results showing that applying successive positive switches to an Erd\H os-R\'enyi graph can significantly increase $\lambda_1$.